Speaking to Japanese gaming site Game Watch (translated), Sony Worldwide Studio President Shuhei Yoshida first explained why the charge exists:

We’re focusing on new services and online, like the “Share” button I previously mentioned and the ability to play using mobile devices with the cloud feature. Our plan is to continue to invest in the future, but investing like this is not compatible with our previous policy that involved reducing costs by every mean to keep the service free.

I apologize for the financial burden on our users, but on PS4 it’ll also provide early access to beta versions, free games and discounts like on PS3 and PS Vita. In addition to that PS Plus members will receive for free a special edition with slightly reduced functions of DriveClub.

He then explained that certain features like asynchronous online gameplay will still be available for the unsubscribed (so, in this case, you’ll presumably be able to play against ghosts of other players’ race performances):

While direct competition in real time won’t be possible [without PS Plus] for instance in the case of DriveClub, asynchronous online gameplay and receiving data from other players will still be free of charge.

You will need a PS Plus subscription to play MMORPGs like Final Fantasy XIV: A Realm Reborn and The Elder Scrolls Online, but that won’t always be the case:

This will also depend on the judgement by the publishers, and there will be some free to play titles that you’ll be able to play without needing a PS Plus subscription, like PlanetSide 2 and DC Universe Online.